Bunuel wrote:
A certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are
A) 0 and 13
B) 0 and 14
C) 1 and 10
D) 1 and 9
E) 2 and 8
We are given that a certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Thus, there were a total of 2 x 32 = 64 classes under this program.
If we let a = the number of teachers teaching one class, b = the number of teachers teaching two classes, and c = the number of teachers teaching 3 classes, we can create the following equations:
a + b + c = 37
a + 2b + 3c = 64
Subtracting equation 1 from equation 2, we have:
(a + 2b + 3c = 64) - (a + b + c = 37)
b + 2c = 27
2c = 27 - b
c = (27 - b)/2
We see that c is the GREATEST when b = 1, and thus (27 - 1)/2 = 26/2 = 13.
We also see that c is the LEAST when b = 27, and thus (27 - 27)/2 = 0/2 = 0.
So, the range of values of n is 0 to 13.
Answer: A
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